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G = D5.D8order 160 = 25·5

The non-split extension by D5 of D8 acting via D8/C8=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C81F5, C401C4, D5.1D8, D10.9D4, D5.1Q16, Dic5.3Q8, C5⋊(C2.D8), C52C86C4, C4⋊F5.4C2, C4.9(C2×F5), C20.9(C2×C4), (C8×D5).3C2, C2.5(C4⋊F5), C10.2(C4⋊C4), (C4×D5).26C22, SmallGroup(160,69)

Series: Derived Chief Lower central Upper central

C1C20 — D5.D8
C1C5C10D10C4×D5C4⋊F5 — D5.D8
C5C10C20 — D5.D8
C1C2C4C8

Generators and relations for D5.D8
 G = < a,b,c,d | a5=b2=c8=1, d2=a-1b, bab=a-1, ac=ca, dad-1=a3, bc=cb, dbd-1=a2b, dcd-1=c-1 >

5C2
5C2
5C4
5C22
20C4
20C4
5C2×C4
5C8
10C2×C4
10C2×C4
4F5
4F5
5C2×C8
5C4⋊C4
5C4⋊C4
2C2×F5
2C2×F5
5C2.D8

Character table of D5.D8

 class 12A2B2C4A4B4C4D4E4F58A8B8C8D1020A20B40A40B40C40D
 size 11552102020202042210104444444
ρ11111111111111111111111    trivial
ρ2111111-11-111-1-1-1-1111-1-1-1-1    linear of order 2
ρ3111111-1-1-1-1111111111111    linear of order 2
ρ41111111-11-11-1-1-1-1111-1-1-1-1    linear of order 2
ρ511-1-11-1ii-i-i111-1-11111111    linear of order 4
ρ611-1-11-1-iii-i1-1-111111-1-1-1-1    linear of order 4
ρ711-1-11-1-i-iii111-1-11111111    linear of order 4
ρ811-1-11-1i-i-ii1-1-111111-1-1-1-1    linear of order 4
ρ92222-2-20000200002-2-20000    orthogonal lifted from D4
ρ102-22-20000002-222-2-20022-2-2    orthogonal lifted from D8
ρ112-22-200000022-2-22-200-2-222    orthogonal lifted from D8
ρ1222-2-2-220000200002-2-20000    symplectic lifted from Q8, Schur index 2
ρ132-2-220000002-22-22-20022-2-2    symplectic lifted from Q16, Schur index 2
ρ142-2-2200000022-22-2-200-2-222    symplectic lifted from Q16, Schur index 2
ρ154400400000-1-4-400-1-1-11111    orthogonal lifted from C2×F5
ρ164400400000-14400-1-1-1-1-1-1-1    orthogonal lifted from F5
ρ174400-400000-10000-111--5-5-5--5    complex lifted from C4⋊F5
ρ184400-400000-10000-111-5--5--5-5    complex lifted from C4⋊F5
ρ194-400000000-1-2222001-5--5ζ83ζ5483ζ5838ζ548ζ5ζ83ζ5383ζ52838ζ538ζ52ζ83ζ5483ζ58ζ548ζ58ζ83ζ5383ζ528ζ538ζ528    complex faithful
ρ204-400000000-122-22001--5-5ζ83ζ5483ζ58ζ548ζ58ζ83ζ5383ζ528ζ538ζ528ζ83ζ5483ζ5838ζ548ζ5ζ83ζ5383ζ52838ζ538ζ52    complex faithful
ρ214-400000000-122-22001-5--5ζ83ζ5383ζ528ζ538ζ528ζ83ζ5483ζ58ζ548ζ58ζ83ζ5383ζ52838ζ538ζ52ζ83ζ5483ζ5838ζ548ζ5    complex faithful
ρ224-400000000-1-2222001--5-5ζ83ζ5383ζ52838ζ538ζ52ζ83ζ5483ζ5838ζ548ζ5ζ83ζ5383ζ528ζ538ζ528ζ83ζ5483ζ58ζ548ζ58    complex faithful

Smallest permutation representation of D5.D8
On 40 points
Generators in S40
(1 26 37 18 10)(2 27 38 19 11)(3 28 39 20 12)(4 29 40 21 13)(5 30 33 22 14)(6 31 34 23 15)(7 32 35 24 16)(8 25 36 17 9)
(1 10)(2 11)(3 12)(4 13)(5 14)(6 15)(7 16)(8 9)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40)
(1 8)(2 7)(3 6)(4 5)(9 18 25 37)(10 17 26 36)(11 24 27 35)(12 23 28 34)(13 22 29 33)(14 21 30 40)(15 20 31 39)(16 19 32 38)

G:=sub<Sym(40)| (1,26,37,18,10)(2,27,38,19,11)(3,28,39,20,12)(4,29,40,21,13)(5,30,33,22,14)(6,31,34,23,15)(7,32,35,24,16)(8,25,36,17,9), (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,9)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40), (1,8)(2,7)(3,6)(4,5)(9,18,25,37)(10,17,26,36)(11,24,27,35)(12,23,28,34)(13,22,29,33)(14,21,30,40)(15,20,31,39)(16,19,32,38)>;

G:=Group( (1,26,37,18,10)(2,27,38,19,11)(3,28,39,20,12)(4,29,40,21,13)(5,30,33,22,14)(6,31,34,23,15)(7,32,35,24,16)(8,25,36,17,9), (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,9)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40), (1,8)(2,7)(3,6)(4,5)(9,18,25,37)(10,17,26,36)(11,24,27,35)(12,23,28,34)(13,22,29,33)(14,21,30,40)(15,20,31,39)(16,19,32,38) );

G=PermutationGroup([(1,26,37,18,10),(2,27,38,19,11),(3,28,39,20,12),(4,29,40,21,13),(5,30,33,22,14),(6,31,34,23,15),(7,32,35,24,16),(8,25,36,17,9)], [(1,10),(2,11),(3,12),(4,13),(5,14),(6,15),(7,16),(8,9),(17,25),(18,26),(19,27),(20,28),(21,29),(22,30),(23,31),(24,32)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40)], [(1,8),(2,7),(3,6),(4,5),(9,18,25,37),(10,17,26,36),(11,24,27,35),(12,23,28,34),(13,22,29,33),(14,21,30,40),(15,20,31,39),(16,19,32,38)])

D5.D8 is a maximal subgroup of
C802C4  C803C4  D5.D16  D5.Q32  (C2×C8)⋊6F5  M4(2)⋊1F5  D8×F5  SD16⋊F5  Q16×F5  Dic5.4Dic6  D5.D24
D5.D8 is a maximal quotient of
C802C4  C803C4  C16.F5  C80.2C4  C401C8  Dic5.13D8  D10.10D8  Dic5.4Dic6  D5.D24

Matrix representation of D5.D8 in GL4(𝔽7) generated by

3133
0163
4422
1560
,
6042
0614
0331
0136
,
3302
1311
2263
3423
,
4133
6266
1212
0210
G:=sub<GL(4,GF(7))| [3,0,4,1,1,1,4,5,3,6,2,6,3,3,2,0],[6,0,0,0,0,6,3,1,4,1,3,3,2,4,1,6],[3,1,2,3,3,3,2,4,0,1,6,2,2,1,3,3],[4,6,1,0,1,2,2,2,3,6,1,1,3,6,2,0] >;

D5.D8 in GAP, Magma, Sage, TeX

D_5.D_8
% in TeX

G:=Group("D5.D8");
// GroupNames label

G:=SmallGroup(160,69);
// by ID

G=gap.SmallGroup(160,69);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,121,151,579,69,2309,1169]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^2=c^8=1,d^2=a^-1*b,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^3,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^-1>;
// generators/relations

Export

Subgroup lattice of D5.D8 in TeX
Character table of D5.D8 in TeX

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